British River Names of The North

Ale Water (Borders) Alne (Northumberland), Ayle Burn (Northumberland) Br. *Alünā, often rendered as Alauna in British Latin, with an uncertain meaning, perhaps from PIE. *ala 'water'.
Blackadder (Borders) Nicolaisen derives this name from *Adarā or *Adarjā from a PIE. root meaning 'water course'.
Calder A common name in Cumbria and Scotland from Br. *caleto- 'hard' and *dubro- 'water' (W. caletwr).
Cam Beck (Cumbria) Br. *cambāco- 'crooked', changed by later association with ON. bekkr 'stream'.
Cocker (Cumbria) Br. *cucrā 'crooked' (G. cuar 'crooked').
Crake (Cumbria) Br. *cracjo- 'rocky' (W. craig 'rock'). Also Crayke in Yorkshire.
Dacre Beck (Cumbria) Br. *dacrū 'tear' (W. deigr).
Derwent (Cumbria) Br. *Derwentjō 'oak river' (W. derw). There was a Roman fort called Derventio at Papcastle. Possibly the Derwenydd mentioned in the early poem Pais Dinogad.
Eden (Cumbria), Eden Water (Borders) Br. *Itunā, written Itouna by Ptolemy. The meaning is obscure, but the name occurs elsewhere in northern England, in Scotland and in Wales as Afon Eden.
Ehen (Cumbria) Uncertain, but the equivalent of C. yeyn, B. yen 'cold' has been suggested.
Ellen (Cumbria) See  Ale Water. The Roman fort of Alauna was situated above the river at Maryport.
Esk (Cumbria) Br. *iscā 'water'. There are several rivers of this name in Cumbria and Scotland.
Gelt (Cumbria) Ekwall refers this to O.I. geilt 'mad, wild' or W. gwyllt (fem. gwellt), which are both from PC. *gʷeltis (: E. wild). The first is probably preferable, since Br. w- does not usually appear as g- in early records.
Glen (Northumberland) From Br. *Glanjo, Glanjā from the stem meaning 'clean, pure, sacred' (W. glan 'clean, pure').
Glenderamackin (Cumbria) See Glenderaterra Beck
Glenderaterra Beck (Cumbria) The first elements are probably Br. *glinnos, glennos 'valley' (W. glyn) and *dubro- 'water' (W. dŵr). The latter elements are obscure.
Irt (Cumbria) Possible British, Ekwall suggests a connection with W. ir 'fresh, green', but also O.E. gyr 'mud'.
Irthing (Cumbria) Ekwall calls this 'a Brit[ish] river-name, but offers no explanation. Early forms are Irthin, Irthing, Erthina and Erthing. Perhaps related to the R. Irt.
Kale Water (Borders) From the Br. *kal- 'hard' (W. caled 'hard').
Kent (Cumbria) Br. *Cunētjō, which occurs elsewhere in England (e.g. the Kennet) and in Wales as Cynwyd.
Leven, Lyne (Cumbria) Br. *limnā 'smooth' (W. llyfn). Another relatively common name in the region.
Lowther (Cumbria) Probably Br. *lowatro-, related to B. laouer 'trough', OI lóathar 'trough' and Gaulish lautro 'bath'.
Lyvennet (Cumbria) Identified with Llwyfenydd, the courtly home of the kings of Rheged in the hengerdd of Taliesin. If this is correct, the origin would be Br. *lēmanjō 'elm river' (W. llwyf, cf. Derwent).
Marron (Cumbria) Ekwall equates the River, recorded as Meran in 1282, with nearby Mockerkin Tarn, which was called Ternmeran in 1343. The latter is apparently an inversion compound with the Br. male name *Mariānos < L. Mariānus (cf. W. Meirion in Meirionydd 'Merioneth').
Mint (Cumbria) A British name from the PIE. root *mim- 'to make a sound'
Mite (Cumbria) Ekwall says this is a British name, but gives no explanation.
Pow Beck (Cumbria) Pow occurs in Cumbria and Scotland and refers to a stream, particularly a slow-moving one. It may come directly from British (cf. W. pwll, C. poll, B. poull 'pool'), but O.E. pōl 'pool' has side forms poll and pull, perhaps influenced by British *pull and Gaelic poll.
Tweed (Borders) An ancient name, perhaps related to Ptolemy's Touésis 'the Spey' which is derived from the PIE. root *tevā 'to swell, be powerful'. There is apparently a PC. root *tew(j)o- 'strength'.  
Wampool (Cumbria) Gospatric's Writ (c.1060) has poll Waðoen, a Celtic inversion compound with O.N. vaðill 'ford'. Poll may be British or Gaelic. 
Whiteadder (East Lothian, Borders) (see Blackadder)
Yarrow Water (Borders), Yarrow (Lancashire) Br. garwo- 'rough' (W. garw).